Download 33-759 Introduction to Mathematical Physics Fall Semester, 2005 Assignment No. 13

33-759 Introduction to Mathematical Physics
Fall Semester, 2005
Assignment No. 13
Due Friday, Dec. 2
READING: Kreyszig: Ch. 1, Secs. 1 2, 3, 5, 9. Chapter 2: take a look at Secs. 1, 2, 3, 7, 8, 9, 10.
We will not be going through this material directly in class, but instead covering the same ground using the
methods of Ch. 3, with some extensions in the Differential Equations Supplement. Chapter 3: Secs. 1, 2, 3,
6. Differential Equations Supplement, Secs. 1 through 6.
READING AHEAD: Kreyszig Ch. 4, Secs. 1 through 6; Ch. 11, Secs. 1, 2, 3, 5, 10.
EXERCISES (available as pdf file at www.andrew.cmu.edu/course/33-759):
1. Turn in at most one page, and not less than half a page, indicating what you have read, examples
or exercises (apart from those assigned below) that you worked out, difficulties you encountered, questions
that came to mind, etc. You may include complaints about the course.
2. Let
(
e−λx
f (x) =
0
Find the convolution
h(y) =
Z
for x > 0,
for x < 0.
∞
f (x)f (y − x) dx,
−∞
of f with itself, both by direct integration and by Fourier inversion of (fˆ(k))2 . [Hint: Pay attention to
whether y is positive or negative.]
3. You should be able to do exercises 2 through 25 at the end of Kreyszig Sec. 1.3, given the hints he
provides. Do not turn anything in.
4. Choose some exercises at random at the end of Kreyszig Sec. 1.5, and make sure you know how to do
them. Do not turn anything in.
5. Riley, Hobson and Bence, Exercise 14.4.
Find the values of α and β that make
α
1
+
dx + (xy β + a) dy
F (x, y) =
x2 + 2
y
an exact differential. For these values solve F (x, y) = 0.
6. Riley, Hobson and Bence, Exercise 14.6. Find an appropriate integrating factor to solve
dy
2x2 + y 2 + x
=−
.
dx
xy
7. Kreyszig Sec. 1.9, problems 1 and 2, slightly modified.
Consider the differential equation xdy/dx = 4y. Are there solutions to the initial value problem with (i)
y(0) = 1, (ii) y(0) = 0? If solutions exist, are they unique? Discuss how the results are related to Kreyszig’s
existence and uniqueness theorems on pp. 53 and 54.
8. OPTIONAL. Kreyszig Sec. 3.1, Nos. 7 and 8.
(continued)
1
9. Relating higher-order linear equations to systems of first-order differential equations.
a) Show that the third-order equation
Ly := y (3) + b2 (x)y (2) + b1 (x)y (1) + b0 (x)y = g(x)
is equivalent to
y0 = Ay + g,
where y1 , y2 , y3 correspond to y, y (1) and y (2) ; A(x) is a suitable 3 × 3 matrix, and g(x) a 3-component
column vector. You should work out the entries of A(x) and g(x) in terms of the quantities in the original
equation.
b) Consider the case of constant coefficients b0 (x) = 2, b1 (x) = −3, b2 (x) = 0. Show that the characteristic equation obtained by inserting the ansatz y = eλx in the homogeneous differential equation Ly = 0 is
(essentially) the same as the characteristic polynomial Det[A − λI] for the matrix A, and find its roots.
c) Find the general solution to the homogeneous equation Ly = 0 considered in (b), and use it to construct
a basis of three solutions yj , j = 1, 2, 3, of the homogeneous system y0 = Ay. Be sure and check that your
yj (x) actually are solutions.
d) Find a basis in which the matrix A for the choice of coefficients in (b) is in Jordan block form. (You
will find that the results in (c) are helpful in this connection.)
10. Find a homogeneous second order linear differential equation for which the functions
y1 (x) = x2 ,
y2 (x) = x2 ln x
are two independent solutions. Find the Wronskian w(x) and show that it satisfies the expected differential
equation. Does the fact that w(x) vanishes at x = 0 contradict the rule that the Wronskian should either
be zero for all x or nonzero for all x? Discuss.
11. Kreyszig Sec. 3.3, Nos. 4 and 10.
12. (Based on Kreyszig No. 6 in Problem Set 3.6.) Consider the system of equations
4 1
y0 = Ay + g where A =
.
2 3
Remember that the inverse of any 2 × 2 matrix has the simple form:
−1
1
d −b
a b
.
=
c d
ad − bc −c a
a) Find two independent solutions to the homogeneous equation, call them y 1 (x) and y2 (x), and form
the 2 × 2 matrix
Ȳ (x) = y1 (x), y2 (x) .
b) Evaluate the Wronskian w(x) = Det[Ȳ (x)], and show that it satisfies the expected differential equation.
c) Obtain Y (x) := exA from Ȳ (x) by finding a constant matrix B such that
Y (x) := exA = Ȳ (x)B.
Verify directly by differentiating its matrix elements that Y satisfies the equation Y 0 = AY .
d) Find the general solution to the inhomogeneous equation when g1 (x) = x, g2 (x) = −x by guessing a
particular solution of the form y = r + xs, solving appropriate equations for the constant vectors r and s,
and then using this as part of the general solution.
e) Find a particular solution to the same inhomogeneous equation by the method of variation of constants,
that is, by using Y −1 (x) or Ȳ −1 (x). Either will work, but one of them is likely to be simpler, so use it rather
than the other.
f) Use the method employed in (e) to obtain a particular solution for the case in which g 1 (x) = e2x and
g2 (x) = e5x . Check by differentiation that the resulting y is a solution of y0 = Ay + g.
2